Strong Law of Large Numbers Application

This was a test question from last semester that I stared at blankly for a long time and couldn't figure out where to begin. Obviously since I need to show convergence almost surely, its the Strong Law of Large Numbers, but beyond that I got really stuck. I'd love some input on at least how to get started. Thanks!

Problem:

Deﬁne the sequence $\displaystyle X_{n}$ inductively by setting $\displaystyle X_{0} = 1$, and selecting $\displaystyle X_{n+1}$ randomly and uniformly from the interval $\displaystyle [0, X_{n}]$. Prove that $\displaystyle n^{-1}$$\displaystyle \log$$\displaystyle X_{n}$ converges almost surely to a constant, and evaluate the limit.

Hint given (I'm having trouble with the /sum so I wrote out the sum expanded):

let $\displaystyle \log$$\displaystyle X_{n}=$$\displaystyle \log$$\displaystyle X_{n} - $$\displaystyle \log$$\displaystyle X_{n-1} + $$\displaystyle \log$$\displaystyle X_{n-1} + ... + $$\displaystyle \log$$\displaystyle X_{1} + $$\displaystyle \log$$\displaystyle X_{0}$

Re: Strong Law of Large Numbers Application

I don't see a SLLN here.

Why don't you obtain the distribution of $\displaystyle X_n$ and then take the log of that rv?

Re: Strong Law of Large Numbers Application

Quote:

Originally Posted by

**puggles** This was a test question from last semester that I stared at blankly for a long time and couldn't figure out where to begin. Obviously since I need to show convergence almost surely, its the Strong Law of Large Numbers, but beyond that I got really stuck. I'd love some input on at least how to get started. Thanks!

Problem:

Deﬁne the sequence $\displaystyle X_{n}$ inductively by setting $\displaystyle X_{0} = 1$, and selecting $\displaystyle X_{n+1}$ randomly and uniformly from the interval $\displaystyle [0, X_{n}]$. Prove that $\displaystyle n^{-1}$$\displaystyle \log$$\displaystyle X_{n}$ converges almost surely to a constant, and evaluate the limit.

Hint given (I'm having trouble with the /sum so I wrote out the sum expanded):

let $\displaystyle \log$$\displaystyle X_{n}=$$\displaystyle \log$$\displaystyle X_{n} - $$\displaystyle \log$$\displaystyle X_{n-1} + $$\displaystyle \log$$\displaystyle X_{n-1} + ... + $$\displaystyle \log$$\displaystyle X_{1} + $$\displaystyle \log$$\displaystyle X_{0}$

With $\displaystyle \ln X_{n}$ You mean $\displaystyle E \{\ln X_{n}\}$... don't You?...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

Re: Strong Law of Large Numbers Application

Quote:

Originally Posted by

**chisigma** With $\displaystyle \ln X_{n}$ You mean $\displaystyle E \{\ln X_{n}\}$... don't You?...

If the answer is 'yes', then the quantity $\displaystyle \mu_{n}= E \{X_{n}\}$ is the solution of the difference equation...

$\displaystyle \mu_{n+1}= \frac{\mu_{n}}{2}\ ,\ \mu_{0}=1$ (1)

... so that is...

$\displaystyle \mu_{n}= \frac{1}{2^{n}}$ (2)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

Re: Strong Law of Large Numbers Application

Quote:

Originally Posted by

**chisigma** If the answer is 'yes', then the quantity $\displaystyle \mu_{n}= E \{X_{n}\}$ is the solution of the difference equation...

$\displaystyle \mu_{n+1}= \frac{\mu_{n}}{2}\ ,\ \mu_{0}=1$ (1)

... so that is...

$\displaystyle \mu_{n}= \frac{1}{2^{n}}$ (2)

... and setting $\displaystyle \lambda_{n}= E \{\ln X_{n}\}$ is...

$\displaystyle \lambda_{n+1}= \frac{1}{X_{n}}\ \int_{0}^{X_{n}} \ln x\ dx = \lambda_{n}-1\ ,\ \lambda_{0}=0$ (1)

... so that is...

$\displaystyle \lambda_{n}= -n$ (2)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

Re: Strong Law of Large Numbers Application

Thanks everyone for your help. Sorry I took so long to respond... the assignment is already turned in, but your comments have helped me understand this much better! Thank you